the picture in the attached figure
we know that
in the right triangle ABD
tan A=BD/AD------>tan A= h/x-----> sin A/cos A=h/x-----> equation 1
in the right triangle ABC
cos C=DC/BC----> cos C=y/a-------> equation 2
sin C=BD/BC-----> sin C=h/a------> equation 3
remember that
angle A +angle C=90 degrees-----> by complementary angles
so
cos A=sin C
sin A=cos C
substitute equation 2 and equation 3 in equation 1
sin A/cos A=h/x-----> (y/a)/(h/a)=h/x-------> (y/h)=(h/x)
then
x/h=h/y
therefore
the term to complete the proportion is h
the answer is
h
